TITLE: Does this series of angles converge?
QUESTION [6 upvotes]: Consider the sequence of squares and angles as in this figure:

Since $\tan \alpha_n=\frac{1}{n}$, we can show that $\alpha_1+\alpha_2+\alpha_3=\frac{\pi}{2}$
(see :Determine the angle of 3 drawn lines from each corner of 3 congruent squares)
For $n>3$ the sum of the angles
$$
\beta_n= \sum_{i=1}^n \alpha_i
$$
 becomes more difficult to find using the tigonometric formulas for the tangent of sum angles. So, the first question is if there is some other method to find this sum.
The second question is if the series
$$
\sum_{i=1}^\infty \alpha_i
$$
converge or not and, if it converges, what is the sum.
I've tempted a numerical experiment that gives: $\beta_{1000}=\pi \cdot2.294981074\ldots$ and $\beta_{2000}=\pi \cdot2,515537077\ldots$
But, obviously, this is not conclusive.

REPLY [2 votes]: $$
\tan \alpha_n = \frac 1 n.
$$
$$
\frac x {\tan x} \to 1 \text{ as }x\to0.
$$
Hence
$$
\frac{\alpha_n}{1/n} = \frac{\alpha_n}{\tan\alpha_n} \to 1 \text{ as }n\to\infty.
$$
If $a_n,b_n>0$ for all $n$ and $\lim\limits_{n\to\infty}\dfrac{a_n}{b_n}$ is a strictly positive number, then the two series $\sum\limits_{n=1}^\infty a_n$ and $\sum\limits_{n=1}^\infty b_n$ either both converge or both diverge.  We know that $\sum\limits_{n=1}^\infty \frac 1 n$ diverges.  Therefore the other series also diverges.